Electron. J. Diff. Equ., Vol. 2010(2010), No. 67, pp. 1-18.

Instability of elliptic equations on compact Riemannian manifolds with non-negative Ricci curvature

Arnaldo S. Nascimento, Alexandre C. Gonçalves

We prove the nonexistence of nonconstant local minimizers for a class of functionals, which typically appear in scalar two-phase field models, over smooth N-dimensional Riemannian manifolds without boundary and non-negative Ricci curvature. Conversely, for a class of surfaces possessing a simple closed geodesic along which the Gauss curvature is negative, we prove the existence of nonconstant local minimizers for the same class of functionals.

Submitted November 15, 2009. Published May 8, 2010.
Math Subject Classifications: 35J20, 58J05.
Key Words: Riemannian manifold; Ricci curvature; local minimizer; Gamma-convergence; reaction-diffusion equations.

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Arnaldo S. Nascimento
Universidade Federal de São Carlos
DM, São Carlos, SP, Brazil
email: arnaldon@dm.ufscar.br
Alexandre C. Gonçalves
Universidade de São Paulo
FFCLRP, Ribeirão Preto, SP, Brazil
email: acasa@ffclrp.usp.br

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